a flute is essentially a pipe open at both ends. the length of a flute is approximately 66.1 cm. what is the third harmonic of a flute when all keys are closed, making the vibrating air column approximately equal to the length of the flute?

To find the third harmonic of a flute, we need to consider the fact that a flute is essentially a pipe open at both ends. In this case, the length of the flute is given as approximately 66.1 cm.

The formula for the length of a resonant air column in a pipe open at both ends to produce the nth harmonic is:

L = (2n-1) * (λ/4)

Where:
L is the length of the vibrating air column
n is the harmonic number
λ (lambda) is the wavelength of the sound wave corresponding to the harmonic

Since we want to find the third harmonic, n = 3.

From the formula, we can rearrange and solve for the wavelength:

(2n-1) * (λ/4) = L

(2*3 - 1) * (λ/4) = 66.1 cm

5 * (λ/4) = 66.1 cm

λ/4 = 66.1 cm / 5

λ/4 = 13.22 cm

Now we can solve for the wavelength:

λ = 13.22 cm * 4

λ = 52.88 cm

Therefore, the wavelength of the third harmonic is approximately 52.88 cm when all keys on a flute are closed, making the vibrating air column length approximately equal to the length of the flute.

To determine the third harmonic of a flute when all keys are closed, you need to consider the relationship between the length of the vibrating air column and the wavelength of the sound produced.

Here's what you need to do step-by-step:

1. Recall that in a closed-end pipe (like a flute), the harmonic series follows the formula:

𝑓𝑛 = 𝑛(𝑣/2𝐿)

Where:
𝑛 represents the harmonic number (e.g., 1st, 2nd, 3rd),
𝑓𝑛 is the frequency of the 𝑛th harmonic,
𝑣 is the speed of sound, and
𝐿 is the length of the air column.

2. Determine the speed of sound. The speed of sound in air is approximately 343 meters per second (m/s), but since you'll be using centimeters for length, you need to convert it to centimeters per second (cm/s). Therefore:

𝑣 = 34300 cm/s

3. Substitute the values into the formula to calculate the frequency of the third harmonic (𝑛 = 3) when all keys are closed:

𝑓3 = 3(34300 cm/s) / (2 * 66.1 cm)

4. Calculate the result:

𝑓3 = 514500 cm/s / 132.2 cm

Simplifying:

𝑓3 ≈ 3890.32 Hz

So, the approximate frequency of the third harmonic of a flute when all keys are closed would be approximately 3,890.32 Hz.

If .661cm is a half wavelength, then lambda = 2*.661m, and third harmonic is 1/3 of that wavelength, or 2*.661/3=.440m

f= speedsound/wavelength= appx 343/.440=778hz check my work.